12th Joint European Thermodynamics Conference
Brescia, July 1-5, 2013
HYBRID ATOMISTIC-CONTINUUM APPROACH TO DESCRIBE INTERFACIAL
PROPERTIES BETWEEN IMMISCIBLE LIQUIDS
K. M. Issa*,1 , and P. Poesio*,2
*Università degli Studi di Brescia, via Branze 38, 25123, Brescia Italy
email:[email protected] , [email protected]
ABSTRACT
The study of immiscible liquid-liquid interfaces (LLIs) is of importance in many phenomena in engineering, chemical, and
biological systems. At an immiscible LLI, a slip occurs as a result of poor mixing, and relatively weaker atomic interactions
between the two liquids. One of the main difficulties in modeling immiscible (and partially miscible) LLIs is the assignment
of boundary conditions at the interface. In continuum-based modeling of macroscale systems, a no-slip boundary condition is
generally assumed at the LLI. The issue of interfacial slip, however, becomes especially relevant for micro-, and nanofluidics
where interface dynamics play a key role, and the slip magnitude can strongly affect the flow behavior. In that respect, molecular
dynamics (MD) is a vital tool for modeling LLIs at the atomic scale. In this paper, we present a hybrid atomistic-continuum
(HAC) approach that utilizes MD at the LLI to directly extract boundary conditions needed by a continuum solver. Our focus is
on the treatment of the atomistic subdomain (ΩA ), specifically when it comes to the proper termination of ΩA , and the coupling
to an external continuum field. The model is tested using a Couette flow under varying flow speeds. Finally, we demonstrate the
ability of the model to accurately predict the slip coefficient at the LLI.
INTRODUCTION
(ΩA ). By limiting ΩA to the LLI, boundary conditions for the
NS solution at the interface are naturally recovered through MD.
With this domain-decomposition approach, the two descriptions
partially overlap in a region (ΩC→A + ΩA→C ) where information is exchanged [5]. The accuracy of the method relies on
the proper application of continuum state variables onto ΩA ,
and the consistency of transport coefficients between the two
descriptions.
In this paper, we present a one-way coupling algorithm for
the HAC modeling of one-dimensional LLIs. The focus is on
the proper termination of ΩA and on the imposition of boundary
conditions of the form ΩC→A . The boundaries of ΩA are modeled as reflective walls supplemented with an adaptive boundary
force, to prevent artificial density layering. The method is tested
using a Couette flow with different velocities. The velocity profiles at the LLI are compared with the continuum NS solution.
The predicted viscosity, and slip coefficients are shown to agree
well with the literature.
Liquid-Liquid Interfaces (LLIs), formed by two immiscible
liquids, occur in a wide range of systems. For instance, in biology, interfaces between two immiscible liquid electrolyte solutions are of great importance as they occur in tissues and cells
of all living organisms. In oil and gas industry, the balance between break-up and coalescence (both interfacial phenomena)
determine the occurrence of phase inversion, a process that can
lead to the blockage of the entire pipeline with a huge economical impact. Furthermore, in recent lab-on-a-chip technology,
liquid droplets, moving through an immiscible liquid, are used
as chemical (and biological) reactors, where reactions are carried out while the droplet is transported along micro-channels.
Hence, the proper understanding, and modeling of LLI dynamics is of great significance to many industries.
The slip behavior at immiscible LLIs is a nanoscale phenomenon, and requires atomic resolution to accurately capture
the relevant physics. Continuum methods generally assume a
no-slip boundary condition at the LLI, which could provide reasonable accuracy on a macroscale level. This approximation
loses validity at the micro- and nanoscale, due to the relatively
higher surface-to-volume ratio. Additionally, the amount of slip
at such scales can be comparable to the characteristic length of
the system and therefore should be carefully accounted for.
Continuum theory, in the form of the Navier-Stokes (NS)
equations, can accurately predict flow dynamics in the regions
far removed from LLI effects. On the other hand, molecular
dynamics (MD) is capable of capturing important nanoscale
physics at the LLI [1, 2, 3, 4]. The method is, however, computationally demanding and modeling is presently limited to systems within the nanoscale. Hybrid atomistic-continuum (HAC)
modeling can alleviate these shortcomings by decomposing the
domain into a continuum description (ΩC ) and an atomistic one
METHODOLOGY
Particles in the atomistic subdomain (ΩA ) interact via the
Lennard-Jones (LJ) potential:
"
ULJ (ri j ) = 4ε
σ
ri j
12
σ
−β
ri j
6 #
(1)
where ri j is the separation distance between particles i, and j.
The energy, and length scales are taken to be those of argon: ε =
0.996 kJmol−1 , and σ = 3.4 Å (1 Å = 10−10 m). The potential
is force-shifted [6] and a cutoff radius of rc = 3σ = 10.2Å is
employed. The parameter β is used to tune the attractive part of
the potential between the two liquids (L1 , and L2 ). To induce
201
complete immiscibility, we assign β11 = β22 = 1.0 and β12 =
0.0. The position (ri ) and velocity (vi ) vectors of each particle i
are governed by Newton’s equation of motion:
dri
= vi
dt
(2)
dvi
Fi
=
dt
mi
(3)
where Fi = − ∑ j6=i ∇ri j ULJ is the total force acting on particle i,
with mass mi = 40 gmol−1 . Eqs. (2)-(3) are solved using the
leap-frog algorithm:
n+1/2
vi
n−1/2
= vi
+ δt Fni /mi
(4)
Figure 1.
Atomistic liquid (L1 + L2 ) subdomain (ΩA ) at the LLI. Ex-
ternal field quantities are applied within the shaded regions denoted by
n+1/2
rn+1
= rni + δt vi
i
j
ΩC→A ( j = 1, 2).
(5)
where δt is the time step. Superscripts in Eqs. (4)-(5) denote
the relative time level of each variable.
The atomistic subdomain ΩA describing the LLI is shown in
Fig. 1. The system dimensions are: Lx × Ly × Lz = 14.2σ ×
13.6σ × 6.8σ. The size is chosen to give a bulk liquid density of
ρ = 1369kgm−3 (ρ∗ = ρσ3 Nav m−1 = 0.81), with a total of N =
1024 particles. Periodic boundary conditions are used along the
y- and z-axes. In the x-direction, the system is terminated using
a reflective wall and an adaptive boundary force (details below).
Velocity and temperature measurements are recorded in bins of
size δx = 0.79σ along the x-axis. To capture the LLI, a higher
bin resolution of δxρ = 0.1δx is used for the density profiles.
The system is equilibrated at T = 132K (T ∗ = kb T /ε = 1.1),
using a Berendsen thermostat [7] with a time constant of 0.5
ps (1 ps = 10−12 s). A time step of δt = 5fs (1 fs = 10−15 s,
δt ∗ = 0.0023t˜, t˜ = σ m0.5 ε−0.5 ) is used with total runs of 15 ns
duration.
Particles that attempt to leave ΩA along the non-periodic xaxis are specularly reflected back into ΩA by reversing the velocity component normal to that plane (vx ). This guarantees
a constant number of particles (N) in ΩA , however, it does
not account for the lack of periodicity along that direction.
Thus, causing artificial density layering normal to the reflective
boundary. One way to alleviate this issue is to supplement the
reflections with a boundary force that ’mimics’ on average the
forces felt by a particle in a periodic system [8]. The boundary
force in [8] was derived by measurement in a periodic system
and applied as a function of distance to the reflective boundary.
For supercritical conditions, this was found to significantly reduce the artificial layering. However, the performance of this
technique was shown to deteriorate at higher densities, and/or
lower temperatures, conditions which are relevant for LLI analysis. This drawback was overcome by extracting the boundary
force using a control algorithm [9, 10], which we employ in our
model. First, the density in each bin is averaged in time intervals
of 5 ps. The noise in the density profile is then reduced by passing it twice through a Gaussian filter as follows (see appendix
A for details):
ρ1 (x) = φ−1
Z
ρ2 (x) = φ−1
Z
h
i
ρ(x) exp − (x − x1 )2 /φ2 dx1
h
i
ρ1 (x) exp − (x − x2 )2 /φ2 dx2
where: ρ1 and ρ2 are the density profiles subsequent to the first
and second Gaussian smoothing, respectively. The integrals are
evaluated discretely with a cutoff of 3δxρ , and φ = 2δxρ . The
gradient of the density profile ∇ρ2 (x) is then used as a correction mechanism to adapt the boundary force Fb which is initially
at zero:
√
Fb (x,t + tad ) = Fb (x,t) − ∇ρ2 λ
(8)
where tad is the 5 ps averaging period. The parameter λ is used
to enact corrections faster away from the boundary which helps
prevent the forces close to the boundary from over-shooting. In
this study, we select λ = 1.1 × 10−4 q, where q is the bin number
counted from the boundary for each liquid, respectively.
The system is first thermostatted for 50 ps, with reflective
boundaries only. The adaptation of Fb commences following
the equilibration period. Additionally, the application of boundj
ary conditions to the particles in ΩC→A ( j = 1, 2) is carried out
simultaneously. For the purpose of studying LLI dynamics in
the presence of a Couette flow, we apply opposing velocities of
uC∗ = ± 0.2, 0.4, 0.6, and 0.8, along the y-axis. The velocity uC∗
is ramped up to its target value over a period of 50 ps. For each
liquid, uC∗ is enforced by applying an additional body force FuC
j
in the y-direction to each atom within ΩC→A ( j = 1, 2), as given
by (see Appendix B):
FuC =
Fcomy
m
uC − vcomy −
δt
NΩ j
(9)
C→A
vcomy =
1
N j
Ω
C→A
∑
NΩ j
vy,i
(10)
i=1
C→A
where vcomy is the center-of-mass y-velocity of particles in
j
ΩC→A ( j = 1, 2), and Fcomy is the corresponding net force. Furthermore, the temperature is maintained at TC∗ = 1.1 via direct
velocity scaling about the center-of-mass velocity:
(6)
s T
C
C→A
TΩ j
vi = vi − uΩ j
(7)
C→A
202
+ uΩ j
C→A
(11)
Figure 2.
∗
Adaptive boundary force (Fb ) for L1 with uC
= 0.2. Distance
Figure 4.
measured from the reflective boundary of Ω1C→A .
∗
Velocity profile across the LLI for uC
= ±0.4, and the analyt-
ical solution given by Eq. (12). Shaded data points are used to measure
apparent slip (δu).
Table 1.
All values are in reduced units. Where appropriate, results are
listed in the form L1 (L2 ).
u∗C
du
dx
(×102 )
τ∗ (×102 )
η∗
δu∗
α
0.2
2.10 (2.11)
4.47 (4.46)
2.13 (2.12)
0.115
2.56
0.4
4.23 (4.20)
9.20 (9.20)
2.17 (2.19)
0.220
2.39
0.6
6.33 (6.38)
13.9 (13.9)
2.20 (2.18)
0.326
2.36
0.8
8.69 (8.66)
18.2 (18.3)
2.13 (2.14)
0.522
2.83
∗ = 0.2, with (+) and without (-) the
Density profiles for uC
action of Fb . The shaded region marks the depleted zone at the LLI.
Figure 3.
D
where uΩ j
C→A
E
= [0, uC , 0] for j = 1, and [0, −uC , 0] for j = 2.
The variable TΩ j
particles in
the velocity profiles, we first compare the viscosity of the liquids
to that measured using equilibrium MD [11], at a representative
state point of ρ∗ = 0.81, and T ∗ = 1.1. This is carried out by
calculating the average amount of momentum (∆p) added every
j
time step in ΩC→A ( j = 1, 2) to maintain the flow. Using ∆p,
the shear stress (τ), or momentum flux, is estimated using:
j
ΩC→A
is the instantaneous temperature of
C→A
( j = 1, 2).
RESULTS & DISCUSSION
τ=
Data sampling was started after 5 ns, for which the boundary
force, and velocity profiles exhibited steady-state behavior. The
collection of data was carried out for an additional 10 ns. The
adaptive boundary force (Fb ) for uC∗ = 0.2 is shown in Fig. 2.
Close to the boundary, Fb is repulsive in nature, with an attractive component further away. The effectiveness of Fb in eliminating the density layering at the boundary is shown in Fig 3.
A uniform density of ρ∗ = 0.81 can be seen away from the depleted region at the LLI between the two immiscible liquids.
The steady-state velocity profile for u∗C = ±0.4 is shown in
Fig. 4, along with the analytical solution for a Couette flow
j
(d 2 u/dx2 = 0). Using the imposed velocities in ΩC→A ( j = 1, 2)
as boundary conditions, the resulting analytical velocity profile
is given by:
u(x) = −
2uC
x + uC
Lx
∆p
δtA
(13)
where A = Ly × Lz is the cross-sectional area parallel to the
flow. The viscosity is then computed from: η = τ (du/dx)−1 .
The calculation of the velocity gradient (du/dx) from the velocity profiles (similar to Fig. 4) is limited to the linear region
(x/Lx ≈ 0.3) away from the depleted zone. The viscosity values
are given in Table 1, and are in good agreement with the value
of η∗ref = 2.18 obtained from [11]. Furthermore, we compare
the slip coefficient (α) predicted using our model with that reported for immiscible LJ liquids αrefaverage = 2.53 [1]. Using the
velocity jump at the LLI, and the measured shear stress τ, the
slip coefficient is estimated by:
α=
(12)
When normalized by uC , all other cases exhibited a similar
trend. The deviation from the analytical solution increases
closer to the LLI, as a result of the interfacial slip. To validate
δu
τ
(14)
where δu is the apparent slip, and is measured using the data
points indicated in Fig. 4. As can be seen in Table 1, the slip
coefficient is accurately predicted within ΩA .
203
CONCLUSION
of 3δx is given by:
We have presented an HAC algorithm for imposing continuum field variables onto an atomistic subdomain (ΩA ) for the
purpose of studying slip at immiscible LLIs. Density layering
at the interface between the two descriptions was eliminated by
the use of an adaptive boundary force. The momentum flux applied and the resulting velocity gradient in the linear zone produce a viscosity that is in close agreement with that reported
in the literature. The method is also shown to accurately capture the slip coefficient (α) at the LLI. Our plans include the
implementation of this algorithm with two-way coupling to the
continuum description (ΩC ).
x+3δx
ρ1 (x) = φ−1
ρ(x1 )e
∑
−
(x−x1 )2
φ2
(15)
δx
x1 =x−3δx
which expands to:
"
−1
ρ(x − 3δx) e
ρ1 (x) = φ
2
− 3δx
φ
2
− δx
φ
ρ(x − δx) e
ρ(x + 2δx) e
NOMENCLATURE
Nav Avogadro’s number
kb Boltzmann constant [m2 kgs−2 K−1 ]
+ ρ(x − 2δx) e
2
− 2δx
φ
2
− δx
φ
+ ρ(x) + ρ(x + δx) e
2
− 2δx
φ
+ ρ(x + 3δx) e
2 #
− 3δx
φ
+
+
δx
using φ = 2 δx:
REFERENCES
2
3 2
1
ρ(x − 3δx) e−( 2 ) + ρ(x − 2δx) e−(1) +
ρ1 (x) =
2
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boundary conditions at the liquid-liquid interface, Phys.
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[2] Y. Hu, X. Zhang, and W. Wang, Boundary conditions at
the liquid-liquid interface in the presence of surfactants,
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[3] G. Galliero, Lennard-Jones fluid-fluid interfaces under
shear, Phys. Rev. E, vol. 81, pp. 0563061-0563067,2010.
[4] Q. Ehlinger, L. Joly, and O. Pierre-Louis, Giant slip at
liquid-liquid interfaces using hydrophobic ball bearings,
Phys. Rev. Lett., vol. 110, pp. 1045041-1045045,2013.
[5] K. M. Mohamed, and A. A. Mohamad, A review of
the development of hybrid atomistic-continuum methods
for dense fluids, Microfluid. Nanofluid., vol. 8, pp. 283302,2010.
[6] M. P. Allen and D. J. Tildesley, Computer simulation of
liquids, Oxford University Press, New York, 1987.
[7] H. J. C. Berendsen, J. P. M. Postma, W. F. van Gunsteren,
A. DiNola, and J. R. Haak, Molecular dynamics with coupling to an external bath, J. Chem. Phys., vol. 81, pp. 36843690,1984.
[8] T. Werder, J. H. Walther, and P. Koumoutsakos, Hybrid
atomistic-continuum method for the simulation of dense
fluid flows, J. Comput. Phys., vol. 205, pp. 373-390,2005.
[9] E. M. Kotsalis, J. H. Walther, and P. Koumoutsakos, Control of density fluctuations in atomistic-continuum simulations of dense liquids, Phys. Rev. E, vol. 76, pp. 01670910167097,2007.
[10] E. M. Kotsalis, J. H. Walther, E. Kaxiras, and P. Koumoutsakos, Control algorithm for multiscale flow simulations of water, Phys. Rev. E, vol. 79, pp. 04570110457014,2009.
[11] R. L. Rowley, and M. M. Painter, Diffusion and viscosity
equations of state for a Lennard-Jones fluid obtained from
molecular dynamics simulations, Int. J. Thermophys., vol.
18, pp. 1109-1121,1997.
1 2
1 2
ρ(x − δx) e−( 2 ) + ρ(x) + ρ(x + δx) e−( 2 ) +
2
−( 32 )
−(1)2
ρ(x + 2δx) e
+ ρ(x + 3δx) e
Normalizing
by half the sum of the density coefficients:
2
1 2
3
2
1
2e−( 2 ) + 2e−(1) + 2e−( 2 ) + 1 = 1.75208, produces the
2
final discrete equation:
ρi1 = 0.0301ρi±3 + 0.1050ρi±2 + 0.2223ρi±1 + 0.2854ρi (16)
where i refers to the bin number. Equation (16) is suitable for
bins with at least three neighbors on either side. That is, for
i = 4 to Nbin − 3. For the rest of the bins, the contribution to the
smoothed value depends on the maximum number of neighboring bins on either side. The derivation is similar to the above,
and can be shown to produce the following complete set:
ρi1 = ρi
ρi1
ρi1
ρi1
i = 1, or Nbin
= 0.3045ρ
i±1
= 0.1117ρ
i±2
= 0.0301ρ
i±3
0.2854ρ
+ 0.3910ρ
i
+ 0.2365ρ
i±1
+ 0.1050ρ
i±2
i = 2, or Nbin − 1
+ 0.3036ρ
i
+ 0.2223ρ
i±1
i
i = 3, or Nbin − 2
+
i = 4 to Nbin − 3
where i denotes the bin number.
Appendix B: Derivation of flow driving body force (FuC )
The following derivation is based on the leap-frog algorithm
Eqs. (4)-(5). For a system of N particles, the center of mass
velocity along any one dimension is given by:
n− 1
vcom2 =
Appendix A: Discrete Gaussian filter equations
1 N n− 12
∑ vi
N i=1
(17)
n = N F n , then we seek
If the total force at step n is: Fcom
∑i=1 i
n
an additional body force FuC to be applied uniformly to each
For clarity, the density bin spacing is labeled δx instead of
δxρ . In discrete form, the Gaussian filer of Eq. (6) with a cutoff
204
particle in order to drive the flow with a velocity uC . By setting
n+ 1
vcom2
when applied uniformly to each particle, that translates to:
= uC :
n− 1
uC = vcom2 +
δt n
δt N n
Fcom +
∑ FuC
Nm
Nm i=1
(18)
FunC =
where Nm is the total mass of the system. Solving for the total
body force gives:
N
∑
i=1
FunC
Nm
n− 12
n
=
uC − vcom − Fcom
δt
(19)
205
m
δt
Fn
n− 1
uC − vcom2 − com
N
(20)